Evidence of Kardar-Parisi-Zhang scaling on a digital quantum simulator

The paper addresses how macroscopic hydrodynamic behavior emerges from microscopic unitary quantum dynamics in many-body systems, focusing on finite-temperature transport in 1D quantum spin chains. Specifically, it studies the high-temperature anomalous transport at the isotropic point of the spin-1/2 XXZ chain, where prior work has indicated superdiffusive behavior and conjectured Kardar-Parisi-Zhang (KPZ) universality. While analogue quantum simulators have shown notable progress, the work aims to demonstrate that noisy, near-term digital quantum simulators can access hydrodynamic scaling exponents in interacting models over meaningful time windows, despite noise and limited depth. The key research questions are: Can a digital quantum device reproduce the KPZ superdiffusive scaling at the isotropic point of the XXZ model using discrete-time (Floquet/Trotterized) dynamics? And can adding an integrability-breaking staggered field restore diffusive scaling? The study leverages pseudo-random initial states to approximate infinite-temperature correlation functions and implements dynamics tailored to hardware constraints on ibmq-montreal.

Recent research has revealed high-temperature spin superdiffusion at the isotropic point of the XXZ model via open-system approaches, with current scaling consistent with KPZ space-time scaling. Numerical studies of infinite-temperature spin autocorrelations support KPZ universality in the isotropic Heisenberg chain, with indications that anomalous scaling persists at finite temperatures. The precise conditions for KPZ emergence are still debated, though integrability appears central, and generalized hydrodynamics has been extended to include anomalous diffusion. Experimental confirmations include neutron scattering in KCuF3 and analogue simulations in ultracold atoms and polariton condensates, which display the predicted superdiffusive exponents. Importantly, Trotterized (discrete-time) versions of the XXZ model have been shown to be integrable and to retain KPZ scaling at the isotropic point, enabling longer-time simulations without Trotter errors on near-term digital hardware. Error mitigation advances have further improved the viability of digital quantum simulations for quantitative dynamics.

Hardware and mapping: Simulations were performed on the 27-qubit ibmq-montreal superconducting device (quantum volume 128). A 1D XXZ chain with open boundary conditions was mapped onto a 21-qubit linear chain on the device based on its connectivity. Site q0 was reserved (untouched by randomization) to host a spin excitation on a homogeneous background. Initial state preparation via pseudo-random circuits: Following the Richter–Pal protocol, a pseudo-random state approximating an infinite-temperature ensemble was generated with a shallow-depth random circuit over the 21-qubit register, leaving q0 untouched. Each layer comprised: - Single-qubit gates: At layer 1, for each qubit q, randomly choose G from {X^{1/2}, Y^{1/2}, Z^{1/2}}. For layers n>1, choose G from {X^{1/2}, Y^{1/2}, T, previous G}. - Entangling step: Apply a pattern of CX gates across the device, alternating between two patterns (A and B) optimized for device connectivity. The number of layers was chosen based on classical simulations of bipartite entanglement growth, confirming rapid saturation near the Page value, indicating sufficient pseudo-randomness. A sample spin density profile was obtained from 30,000 shots to validate the preparation. Observable and typicality approximation: The target observable is the site-resolved spin autocorrelation C_R(t) = Tr(S^z_R S^z_R(t))/2^L at infinite temperature. Using typicality, this is approximated by C_R(t) ≈ ⟨Ψ| S^z_R S^z_R(t) |Ψ⟩ with errors O(2^{-L}), where |Ψ⟩ is the prepared pseudo-random state. This approach enables evaluation of thermal correlators from a single state on quantum hardware. Model and discrete-time dynamics: The spin-1/2 XXZ Hamiltonian is H_XXZ = Σ_i (S^x_i S^x_{i+1} + S^y_i S^y_{i+1} + Δ S^z_i S^z_{i+1}) with Δ=1 (isotropic). Terms are grouped into disjoint two-site sums H1 (odd bonds) and H2 (even bonds). Discrete-time evolution is implemented via first-order Trotter steps: U(τ) = U_odd(τ) U_even(τ), iterated n times, U(nτ) = [U_odd(τ) U_even(τ)]^n. Two-qubit unitary blocks U_jk(τ) = exp(-i h_{jk} τ) were compiled into device-native gate sequences (based on known optimal two-qubit decompositions) appropriate for the transmon architecture. Floquet perspective and integrability breaking: Rather than approximating continuous time, the Trotterized model is treated as a Floquet system with kicking period τ, described effectively by H(t) = H1 + τ Σ_n H2 δ(t - nτ), which retains KPZ-like scaling in discrete time and avoids Trotter error accumulation. To study diffusion restoration, a staggered field term H3 = Σ_i (-1)^i J^z_i was added, yielding U(τ) = U_even(τ) U_3(τ) U_odd(τ), with U_3(τ) implemented via single-qubit rotations. This explicitly breaks integrability and is expected to yield diffusive transport at high temperatures. Time stepping, weaving, and error mitigation: The optimal Trotter step used on hardware was τ = 4/J, chosen to balance hardware coherence and gate errors while avoiding known many-body resonances. To increase temporal resolution for fitting power laws, a weaving technique was used: prepend shorter-time evolutions U(τ′<τ) as modified initial conditions (τ′ = 1/J, 1.5/J, 2/J), then interleave (“weave”) these trajectories with the main evolution to produce denser effective sampling in time. Zero-noise extrapolation (ZNE) was also tested but did not significantly improve results at these depths and timescales for first-order Trotter steps. Classical benchmarks: Classical simulations of the discrete-time dynamics were performed for both clean and staggered-field cases across multiple Trotter steps to confirm that the long-time power-law decay exponents of C_00(t) are step-size independent (within a safe regime away from resonance conditions). These classical results served as references to identify time windows on hardware where quantum results are reliable and scaling holds. Device performance and calibration: During data collection, qubit-0 readout error was ~2.26%; the average CNOT error (excluding an outlier at 18.4%) across used couplers was ~1.12% (std 0.52%). Sampling noise was negligible compared to device errors; reported error bars focus on device performance. Power-law exponents were obtained by least-squares fits over time windows where classical results show scaling and quantum-classical agreement is good.

- The digital quantum simulation on ibmq-montreal reproduces hydrodynamic scaling of the spin autocorrelation function at high temperature over approximately two decades in time for a 21-qubit XXZ chain at Δ=1. - Clean (integrable) case: The fitted decay exponent from quantum data is a ≈ -0.644, closely tracking the expected KPZ value of -2/3 (relative error ~3.40%). - Staggered-field (integrability-broken) case: The fitted decay exponent is a ≈ 0.505, closely matching the expected diffusive scaling magnitude 1/2 (reported relative error ~1.00%). - Classical discrete-time simulations confirm that the long-time power-law exponents are independent of the Trotter step within the chosen safe range, for both clean and staggered-field models. - On hardware, results align well with classical references up to two decades in time, sufficient to observe the onset of hydrodynamic scaling. The weaving technique increased the number of effective time points, enabling robust power-law fitting. - Zero-noise extrapolation provided limited benefit for the first-order Trotter circuits at these timescales; device errors dominated over sampling noise. - Representative device metrics during runs: qubit-0 readout error ~2.26%; average CNOT error ~1.12% (excluding one 18.4% outlier), std ~0.52%.

The study demonstrates that near-term digital quantum devices can capture emergent hydrodynamic behavior in interacting quantum models when discrete-time (Floquet) dynamics and pseudo-random initial states are used. The pseudo-typical initial state effectively emulates infinite-temperature ensembles and appears resilient to certain noise channels, potentially due to local equivalence to identity and robustness to unital noise like dephasing. Treating the Trotterized dynamics as an exactly defined discrete-time model avoids Trotter errors, enabling longer accessible times for scaling analysis on noisy hardware. The extracted exponents in both integrable and non-integrable regimes are consistent with theoretical expectations and prior analogue experiments, indicating that key transport properties—KPZ superdiffusion and its suppression to diffusion under integrability breaking—are accessible digitally. This constitutes, to the authors’ knowledge, the first extraction of transport exponents of an interacting quantum system on a digital quantum device, and suggests that as hardware quality and qubit counts improve, digital simulators will increasingly probe transport regimes beyond classical reach.

This work provides evidence that a noisy, near-term digital quantum simulator can reproduce KPZ superdiffusive scaling at the isotropic point of the XXZ chain and demonstrate restoration of diffusion upon integrability breaking via a staggered field. By leveraging pseudo-random initial states and discrete-time dynamics mapped to hardware-efficient circuits on a 21-qubit ibmq-montreal device, the study observes power-law decay exponents over two decades in time that agree closely with theoretical expectations and classical benchmarks. The approach showcases how careful initial-state preparation, weaving for enhanced temporal resolution, and treating Trotterized dynamics as a Floquet model can overcome typical NISQ limitations to access hydrodynamic transport signatures. Future work should explore larger systems, deeper circuits with improved error mitigation and calibration, the interplay of noise with pseudo-typical states, and transport in non-integrable models and regimes not tractable via classical numerics.

- Hardware noise and calibration variability limit circuit depth and accessible evolution times; while two decades in time were achieved, longer times for more precise exponent extraction remain challenging. - System size is constrained to 21 qubits with open boundaries, which can introduce finite-size effects in long-time tails of correlation functions. - The pseudo-random initial states deviate from exact Haar randomness; although typicality is robust, residual biases may affect precise quantitative estimates. - Zero-noise extrapolation had limited efficacy for the circuits and timescales studied; other mitigation techniques might yield further improvements. - One CNOT link exhibited a high error rate (18.4% outlier), necessitating careful routing; device-specific mapping may not generalize across hardware. - The analysis relies on matching quantum data to classical benchmarks to select fitting windows; in regimes beyond classical reach, objective window selection would be more difficult.